Question

A ball of mass $m$ moving with a velocity of $v$ made a head on collision with an identical ball at rest. The kinetic energy at the balls after the collision is $\dfrac{3}{4}th$ of the original. What is the coefficient of restitution?
A.)$\dfrac{1}{{\sqrt 3 }}$
B.)$\dfrac{1}{{\sqrt 2 }}$
C.)$\sqrt 2 $
D.)$\sqrt 3 $

Answer 1

Hint- When there is a situation like the masses of both bodies are the same and initially one of them is at rest, then we have some conditions related to the velocity and coefficient of restitution. With the help of that relation and with the help of the given condition we can go for the solution of this problem.

Step By Step Answer:
Here given,
Mass of the both balls =$m$, since both balls are identical.
Before collision,

Initial Velocity of ball 1=$v$
Initial velocity of ball 2=$0$
After collision,

Final velocity of ball 1=${v_2}$
Final velocity of ball 2=$v{'_1}$

Given condition,

kinetic energy at the balls after the collision is $\dfrac{3}{4}th$of the original.
So, here we see that,

${m_1} = {m_2} = m$

And initially the ball 2 is at rest with which the ball 1 is going to collide.
So initially, ${v_{ball2}} = 0$

For this type of condition,

$ev = {v_1} - {v_2}$,

Where $e = $coefficient of restitution.
And also, for the given velocities in the diagram,

$v{'_1} = \left( {\dfrac{{1 + e}}{2}} \right)v$ and ${v_2} = \left( {\dfrac{{1 - e}}{2}} \right)v$
Where \[{v_2}\]=final velocity of ball 1
And $v{'_1}$= final velocity of ball 2.

Now we will proceed with the given condition of the kinetic energy,

${K_f} = \dfrac{3}{4}{K_i}$,
Where ${K_f}$= final kinetic energy
And ${K_i}$=initial kinetic velocity

So $\dfrac{1}{2}mv{'_1}^2 + \dfrac{1}{2}m{v_2}^2 = \dfrac{3}{4}{(\dfrac{1}{2}mv)^2}$
Substituting the values,

${\left( {\dfrac{{1 + e}}{2}} \right)^2} + {\left( {\dfrac{{1 - e}}{2}} \right)^2} = \dfrac{3}{4}$
$ \Rightarrow {(1 + e)^2} + {(1 - e)^2} = 3$
$ \Rightarrow 2 + 2{e^2} = 3$
$ \Rightarrow 2{e^2} = 1$
$ \Rightarrow {e^2} = \dfrac{1}{2}$
$ \Rightarrow e = \dfrac{1}{{\sqrt 2 }}$
So, like this we have found the solution.

Hence the coefficient of restitution for the above given problem is $\dfrac{1}{{\sqrt 2 }}$.
So, option (B) is the correct answer.

Note- This question is from the topic types of collision from the lesson work, power and energy.
There are basically two types of collision between two bodies (regarding the nature of collision).First is head-on collision (here just before the collision, the direction of velocity of each body will be in the line of impact ) and the second is non-head-on collision ( here, the velocity just before the collision is not in the line of impact).